Uncertainty Quantification for Complex Aero-mechanical Systems S - PowerPoint PPT Presentation

Uncertainty Quantification for Complex Aero-mechanical Systems S Adhikari School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/ ∼ adhikaris RAeS, London, 15 May 2008 UQ of complex systems – p.1/58

Outline of the presentation Uncertainty Quantification (UQ) in structural dynamics Review of current approaches Non-parametric approach: Wishart random matrices Parameter selection Computational method Analytical method Parametric approach: Gaussian emulator Frequency response function emulation Random field generation Experimental results Conclusions & future directions RAeS, London, 15 May 2008 UQ of complex systems – p.2/58

Complex aerospace system Complex aerospace system can have millions of degrees of freedom and signifi- cant uncertainty in its numerical (Finite Element) model RAeS, London, 15 May 2008 UQ of complex systems – p.3/58

Sources of uncertainty (a) parametric uncertainty - e.g., uncertainty in geometric parameters, friction coefficient, strength of the materials involved; (b) model inadequacy - arising from the lack of scientific knowledge about the model which is a-priori unknown; (c) experimental error - uncertain and unknown error percolate into the model when they are calibrated against experimental results; (d) computational uncertainty - e.g, machine precession, error tolerance and the so called ‘h’ and ‘p’ refinements in finite element analysis, and (e) model uncertainty - genuine randomness in the model such as uncertainty in the position and velocity in quantum mechanics, deterministic chaos. RAeS, London, 15 May 2008 UQ of complex systems – p.4/58

Structural dynamics The equation of motion: M ¨ q ( t ) + C ˙ q ( t ) + Kq ( t ) = f ( t ) (1) Due to the presence of (parametric/nonparametric or both) uncertainty M , C and K become random matrices. The main objectives in the ‘forward problem’ are: to quantify uncertainties in the system matrices to predict the variability in the response vector q Probabilistic solution of this problem is expected to have more credibility compared to a deterministic solution RAeS, London, 15 May 2008 UQ of complex systems – p.5/58

Current UQ approaches - 1 Two different approaches are currently available Parametric approaches : Such as the Stochastic Finite Element Method (SFEM): aim to characterize parametric uncertainty (type ‘a’) assumes that stochastic fields describing parametric uncertainties are known in details suitable for low-frequency dynamic applications RAeS, London, 15 May 2008 UQ of complex systems – p.6/58

Current UQ approaches - 2 Nonparametric approaches : Such as the Statistical Energy Analysis (SEA) and Wishart random matrix theory: aim to characterize nonparametric uncertainty (types ‘b’ - ‘e’) does not consider parametric uncertainties in details suitable for high/mid-frequency dynamic applications extensive works over the past decade → general purpose commercial software is now available RAeS, London, 15 May 2008 UQ of complex systems – p.7/58

UQ approaches: challenges The main difficulties are due to: the computational time can be prohibitively high compared to a deterministic analysis for real problems, the volume of input data can be unrealistic to obtain for a credible probabilistic analysis, the predictive accuracy can be poor if considerable resources are not spend on the previous two items, and as the state-of-the art methodology stands now (such as the Stochastic Finite Element Method), only very few highly trained professionals (such as those with PhDs) can even attempt to apply the complex concepts (e.g., random fields) and methodologies to real-life problems. RAeS, London, 15 May 2008 UQ of complex systems – p.8/58

Main objectives Our work is aimed at developing methodologies [the 10-10-10 challenge] with the ambition that they should: not take more than 10 times the computational time required for the corresponding deterministic approach; result a predictive accuracy within 10% of direct Monte Carlo Simulation (MCS); use no more than 10 times of input data needed for the corresponding deterministic approach; and enable ‘normal’ engineering graduates to perform probabilistic structural dynamic analyses with a reasonable amount of training. RAeS, London, 15 May 2008 UQ of complex systems – p.9/58

Wishart random matrix approach The probability density function of the mass ( M ), damping ( C ) and stiffness ( K ) matrices should be such that they are symmetric and non-negative matrices. Wishart random matrix (a non-Gaussian matrix) is the simplest mathematical model which can satisfy these two criteria: [ M , C , K ] ≡ G ∼ W n ( p, Σ ) . Suppose we ‘know’ (e.g, by measurement or stochastic modeling) the mean ( G 0 ) and the (normalized) standard deviation ( σ G ) of the system matrices: � � � G − E [ G ] � 2 E F σ 2 G = . (2) � E [ G ] � 2 F RAeS, London, 15 May 2008 UQ of complex systems – p.10/58

Wishart parameter selection - 1 The parameters p and Σ can be obtained based on what criteria we select. We investigate four possible choices. 1. Criteria 1: E [ G ] = G 0 and σ G = � σ G which results p = n + 1 + θ and Σ = G 0 /p (3) σ 2 where θ = (1 + β ) / � G − ( n + 1) and � 2 � β = { Trace ( G 0 ) } 2 / Trace . G 0 � G − 1 �� � − 1 − E � G 0 � 2. Criteria 2: � G 0 − E [ G ] � F and F are minimum and σ G = � σ G . This results: p = n + 1 + θ and Σ = G 0 /α (4) � where α = θ ( n + 1 + θ ) . RAeS, London, 15 May 2008 UQ of complex systems – p.11/58

Wishart parameter selection - 2 � G − 1 � − 1 and σ G = � 1. Criteria 3: E = G 0 σ G . This results: p = n + 1 + θ and Σ = G 0 /θ (5) 2. Criteria 4: The mean of the eigenvalues of the distribution is same as the ‘measured’ eigenvalues of the mean matrix and the (normalized) standard deviation is same as the measured standard deviation: � M − 1 � − 1 , E [ K ] = K 0 , σ M = � E = M 0 σ M and σ K = � σ K . (6) RAeS, London, 15 May 2008 UQ of complex systems – p.12/58

A cantilever plate: front view The test rig for the cantilever plate; front view. RAeS, London, 15 May 2008 UQ of complex systems – p.13/58

A cantilever plate: side view The test rig for the cantilever plate; side view. RAeS, London, 15 May 2008 UQ of complex systems – p.14/58

Physical properties Plate Properties Numerical values Length ( L x ) 998 mm Width ( L y ) 530 mm Thickness ( t h ) 3.0 mm 7860 kg/m 3 Mass density ( ρ ) 2 . 0 × 10 5 MPa Young’s modulus ( E ) Poisson’s ratio ( µ ) 0.3 Total weight 12.47 kg Material and geometric properties of the cantilever plate considered for the experiment. The data presented here are available from http://engweb.swan.ac.uk/ ∼ adhikaris/uq/. RAeS, London, 15 May 2008 UQ of complex systems – p.15/58

Mean of cross-FRF −60 RMT−1 RMT−2 RMT−3 −80 Mean of amplitude (dB) of FRF at point 1 RMT−4 SFEM −100 −120 −140 −160 −180 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Mean of the amplitude of the response of the cross-FRF of the plate, n = 1200 , σ M = 0 . 1326 and σ K = 0 . 3335 . RAeS, London, 15 May 2008 UQ of complex systems – p.16/58

Error in the mean of cross-FRF 20 RMT−1 RMT−2 18 RMT−3 Error in mean of amplitude of FRF at point 1 RMT−4 16 14 12 10 8 6 4 2 0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Error in the mean of the amplitude of the response of the cross-FRF of the plate, n = 1200 , σ M = 0 . 1326 and σ K = 0 . 3335 . RAeS, London, 15 May 2008 UQ of complex systems – p.17/58

Standard deviation of driving-point-FRF −60 RMT−1 RMT−2 Standard deviation of amplitude of FRF at point 2 RMT−3 −80 RMT−4 SFEM −100 −120 −140 −160 −180 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Standard deviation of the amplitude of the response of the driving-point-FRF of the plate, n = 1200 , σ M = 0 . 1326 and σ K = 0 . 3335 . RAeS, London, 15 May 2008 UQ of complex systems – p.18/58

Error in the standard deviation of driving-point-FRF 20 RMT−1 Error in standard deviation of amplitude of FRF at point 2 RMT−2 18 RMT−3 RMT−4 16 14 12 10 8 6 4 2 0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Error in the standard deviation of the amplitude of the response of the driving- point-FRF of the plate, n = 1200 , σ M = 0 . 1326 and σ K = 0 . 3335 . RAeS, London, 15 May 2008 UQ of complex systems – p.19/58

Main observations Error in the low frequency region is higher than that in the higher frequencies In the high frequency region all methods are similar Overall, parameter selection 3 performs best; especially in the low frequency region. RAeS, London, 15 May 2008 UQ of complex systems – p.20/58

Standard deviation: low frequency −60 RMT−1 RMT−2 Standard deviation of amplitude of FRF at point 2 RMT−3 −80 RMT−4 SFEM −100 −120 −140 −160 −180 0 100 200 300 400 500 600 700 800 Frequency (Hz) Standard deviation of the amplitude of the response of the driving-point-FRF of the plate in the low frequency region, n = 1200 , σ M = 0 . 1326 and σ K = 0 . 3335 . RAeS, London, 15 May 2008 UQ of complex systems – p.21/58